In a method of recognizing and tracking a spatial point as described in U.S. patent application Ser. No. 12,047,159, an embodiment is used for illustrating the method of the patent application. In the method, vertically and horizontally arranged 1D optical lenses are used for computing the coordinates of the position of a point light source (or an object point) according to the position of a line image and a related convergent parameters, but the method is not applicable for a plurality of point light sources arranged in a specific space as illustrated in the following examples.
In an 1D vertical focusing lens 1 (which is represented by a short double arrow headed line in FIG. 1(a), and the arrow direction represents the focusing direction of the 1D optical lens) as shown in FIG. 1(a), the line image positions of point light sources o1, o2 disposed at different vertical positions are iy1, iy2 respectively, such that the 1D optical lens 1 can analyze and recognize the point light sources o1, o2 in the vertical direction. However, the same line image position iy1 is obtained when the point light sources o1, o1′ are both disposed on a plane of Z=C (which is a plane perpendicular to the optical axis Z) and situated on the same horizontal line. In other words, the 1D optical lens 1 cannot analyze and recognize the point light sources o1, o1′ in the horizontal direction.
For the 1D horizontal focusing lens 2 as shown in FIG. 1(b), the point light sources o1, o2 are disposed at different horizontal positions and their line images positions are ix1, ix2 respectively, such that the 1D optical lens 2 can analyze and recognize the point light sources o1, o2 in the horizontal direction. However, the same line image position ix1 is obtained when the point light sources o1, o1′ of the 1D horizontal focusing lens 2 are both disposed on a plane of Z=C (which is a plane perpendicular to the optical axis Z) and situated on the same vertical line. In other words, the 1D optical lens 2 cannot analyze and recognize the point light sources o1, o1′ in the vertical direction. Therefore, if the plurality of point light sources disposed at a plane perpendicular to the optical axis (hereinafter referred to as an optical axis perpendicular plane) are arranged at positions perpendicular to the focusing direction, the images will be superimposed, and the spatial positions of the plurality of point light sources cannot be recognized.
As described in the aforementioned patent, three 1D optical lens arranged in a fixed space are used, and if any two or more point light sources are disposed at the optical axis perpendicular plane of any 1D optical lens and arranged at positions perpendicular to the focusing direction of that 1D optical lens, the 1D optical lens will lose the recognition function. This result can be shown clearly by the following theoretical analysis.
Refer to FIG. 2(a) for the schematic view of the principle of imaging by a 1D vertical focusing lens.
After an object point of a point light source located at P(0,YP, ZP) forms a line image at the position I(0,yi,0) by the 1D optical lens 3, the relation between positions of the object point and the line image follows the principle of geometric optical imaging as shown in the following equation.
                                          1                          l              o                                +                      1                          l              i                                      =                  1          f                                    (        1        )            
Where lo is the object distance of the point light source P(0,Yp,Zp), li is the image distance, and f is the focal length of the 1D optical lens 3. In the theory of geometric optical imaging, a non-deviated light exists between the point light source P(0,Yp,Zp) and the image point I(0,yi, 0), and the light passes through a geometric center Olens of the 1D optical lens 3. If the object distance lo is much greater than the image distance li, or lo>>li, then the relation of li=f can be obtained.
Refer to FIG. 2(b) for the schematic view of the imaging characteristics of a 1D vertical focusing lens.
For a point light source arbitrarily disposed at P(0,Yp,Zp), a transverse line image is formed by the 1D vertical focusing lens 3 and situated at a position I(0,yi,0) For another arbitrary point light source P(Xp,Yp,ZP) situated in the same horizontal direction, the formed image is also a transverse line and situated at the same position I(0,yi,0). Therefore, P(0,YP,ZP) is defined as an axial point light source, and P(Xp,Yp,Zp) is defined as a conjugated point light source of P(0,Yp, Zp).
Refer to FIG. 2(c) for a schematic view of the characteristics of imaging of a 1D optical lens in arbitrary focusing direction.
As to the coordinate system O(X,Y,Z), the focusing direction of the 1D focusing lens is rotated at an angle θ with respect to axis Z. A new coordinate system O1(x,y, z) superimposed on the coordinate system O(X,Y,Z) is defined, such that the x−y axes are also rotated at an angle θ with respect to axis Z. Therefore, in the new coordinate system O1(x,y,z), let P1(0,yp,zp) be an axial point light source and P1(xp,Yp,zp) be a conjugated point light source of P1(0,yp, zp) In the coordinate system O(X,Y,Z), the coordinate of P1(xp,yp,zp) is P(Xp,Yp,Zp).
Refer to FIG. 2(d) for a schematic view of a 1D optical lens arranged arbitrarily in the space.
In the world coordinate system O(X,Y,Z), also named as a visual space coordinate system, the point light source Pi is disposed at a position (Xi,Yi,Zi), where 1≦i≦N and N is any integer, and the coordinates (Xi,Yi,Zi) of the point light source Pi are also named as object point coordinates. As to the coordinates of all point light sources, they are called object point group coordinates and defined as the center coordinates of an object point group as follows:
                                          X            C                    =                                                    ∑                                                      i                    =                                                                                  ⁢                    1                                    ,                                                                          ⁢                  N                                            ⁢                                                          ⁢                              X                i                                      N                          ;                              Y            C                    =                                                    ∑                                                      i                    =                                                                                  ⁢                    1                                    ,                                                                          ⁢                  N                                            ⁢                                                          ⁢                              Y                i                                      N                          ;                              Y            C                    =                                                    ∑                                                      i                    =                                                                                  ⁢                    1                                    ,                                                                          ⁢                  N                                            ⁢                                                          ⁢                              Z                i                                      N                                              (        2        )            
The Z-axis of the world coordinate system is rotated at an angle Θ with respect to Y-axis first, then is rotated at an angle Φ with respect to X-axis, wherein the positive and negative values of the angle are defined according to the right hand rule. Therefore, the rotated world coordinate system can be defined as O″(X″,Y″,Z″). Further, several other image coordinate systems Oj″(Xj″,Yj″Zj″) can be defined, so that the origin of the image coordinate system Oj″(Xj″,Yj″,Zj″) is situated at the position (hxj,hyj,hzj) of the world coordinate system O″(X″,Y″,Z″). For simplicity, FIG. 2(d) only shows the components of hxj. Further, a 1D vertical focusing lens Lj is set on a Z″j axis of the image coordinate system Oj″(Xj″,Yj″,Zj″) and at a position fj from the origin of the image coordinate system, wherein Fj is the geometric center of the 1D optical lens Lj, and fj is the focal length. Further, the image coordinate system Oj″(Xj″,Yj″, Zj″) is rotated at an angle θj with respect to the Yj″ axis first, and then is rotated at an angle ρj with respect to the Z″ j axis, wherein the positive and negative values of the angle are defined according to the right hand rule. Therefore, the rotated image coordinate system can be defined as Oj″(Xj″,Yj″,Zj″). Let the object distance of the point light source Pi be much greater than the focal length fi, and the plane of the focal point becomes an image plane situated on the plane Xj″−Yj″ and Zj″=0 of the image coordinate system Oj″(Xj″,Yj″,Zj″). In the world coordinate system O″(X″,Y″,Z″),the point light source Pi is situated at the position Pi″(Xi″,Yi″,Zi″), and in the image coordinate system Oj″(Xj″,Yj″,Zj″) the point light source Pi situated at the position Pij(xoij,yoij,zoij). In the image coordinate system Oj″(Xj″,Yj″,Zj″), let the point light source Pij(xoij,yoij,zoij) be the conjugated point light source and Pij(0,Yoij,zoij) be the axial point light source. Then the line image position of Pij(0,yoij,zoij) is situated at Iij(0,ysij,0), and their geometric optical relation is given below:
                              y          oij                =                              -                                                            z                  oij                                -                                  f                  j                                                            f                j                                              ⁢                      y            sij                                              (        3        )            
According to the relation of coordinate transformation between the image coordinate system Oj″(Xj″,Yj″,Zj″) and the world coordinate system O(X,Y,Z) and the spatial geometric arrangement of the point light sources in the world coordinate system O(X,Y,Z), the necessary quantity of 1D optical lenses Lj can be derived and the coordinate (Xi,Yi,Zi) of each point light source Pi in the world coordinate system O(X,Y,Z) can be calculated. The derivation and calculation are discussed as follows:
The relation of coordinate transformation between the image coordinate system Oj″(Xj″,Yj″,Zj″) and the world coordinate system O(X,Y,Z) is given below:
                                          (                                                                                X                    i                                                                                                                    Y                    i                                                                                                                    Z                    i                                                                        )                    =                                                                      R                  j                                ⁡                                  (                                      Θ                    ,                    Φ                    ,                                          θ                      j                                        ,                                          ρ                      j                                                        )                                            ⁢                              (                                                                                                    x                        oij                                                                                                                                                y                        oij                                                                                                                                                z                        oij                                                                                            )                                      +                          (                                                                                          h                      xj                                                                                                                                  h                      yj                                                                                                                                  h                      zj                                                                                  )                                      ⁢                                  ⁢        Where                            (        4        )                                                                    R              j                        ⁡                          (                              Θ                ,                Φ                ,                                  θ                  j                                ,                                  ρ                  j                                            )                                =                      (                                                                                R                                          j                      ⁢                                                                                          ⁢                      11                                                                                                            R                                          j                      ⁢                                                                                          ⁢                      12                                                                                                            R                                          j                      ⁢                                                                                          ⁢                      13                                                                                                                                        R                                          j                      ⁢                                                                                          ⁢                      21                                                                                                            R                                          j                      ⁢                                                                                          ⁢                      22                                                                                                            R                                          j                      ⁢                                                                                          ⁢                      23                                                                                                                                        R                                          j                      ⁢                                                                                          ⁢                      31                                                                                                            R                                          j                      ⁢                                                                                          ⁢                      32                                                                                                            R                                          j                      ⁢                                                                                          ⁢                      33                                                                                            )                          ⁢                                  ⁢        and                            (        5        )                                                      R                          j              ⁢                                                          ⁢              l              ⁢                                                          ⁢              m                                ≡                      f            ⁡                          (                              Θ                ,                Φ                ,                                  θ                  j                                ,                                  ρ                  j                                            )                                      ,                  1          ≤          l          ≤          3                ,                  1          ≤          m          ≤          3                                    (        6        )            
Rjlm is a function of Θ,Φ,θj, ρj. With the matrix operation, Pij(xoij,yoij,zoij) can be computed as follows:
                                          (                                                                                x                    oij                                                                                                                    y                    oij                                                                                                                    z                    oij                                                                        )                    =                                                    r                j                            ⁡                              (                                  Θ                  ,                  Φ                  ,                                      θ                    j                                    ,                                      ρ                    j                                                  )                                      ⁡                          [                                                (                                                                                                              X                          i                                                                                                                                                              Y                          i                                                                                                                                                              Z                          i                                                                                                      )                                -                                  (                                                                                                              h                          xj                                                                                                                                                              h                          yj                                                                                                                                                              h                          zj                                                                                                      )                                            ]                                      ⁢                                  ⁢                  where          ,                                    (        7        )                                                                                                      r                  j                                ⁡                                  (                                      Θ                    ,                    Φ                    ,                                          θ                      j                                        ,                                          φ                      j                                                        )                                            =                            ⁢                                                                    R                    j                                    ⁡                                      (                                          Θ                      ,                      Φ                      ,                                              θ                        j                                            ,                                              φ                        j                                                              )                                                                    -                  1                                                                                                        =                            ⁢                                                (                                                                                                              R                                                      j                            ⁢                                                                                                                  ⁢                            11                                                                                                                                                R                                                      j                            ⁢                                                                                                                  ⁢                            12                                                                                                                                                R                                                      j                            ⁢                                                                                                                  ⁢                            13                                                                                                                                                                                        R                                                      j                            ⁢                                                                                                                  ⁢                            21                                                                                                                                                R                                                      j                            ⁢                                                                                                                  ⁢                            22                                                                                                                                                R                                                      j                            ⁢                                                                                                                  ⁢                            23                                                                                                                                                                                        R                                                      j                            ⁢                                                                                                                  ⁢                            31                                                                                                                                                R                                                      j                            ⁢                                                                                                                  ⁢                            32                                                                                                                                                R                                                      j                            ⁢                                                                                                                  ⁢                            33                                                                                                                                )                                                  -                  1                                                                                                                        =                                ⁢                                  (                                                                                                              r                                                      j                            ⁢                                                                                                                  ⁢                            11                                                                                                                                                r                                                      j                            ⁢                                                                                                                  ⁢                            12                                                                                                                                                r                                                      j                            ⁢                                                                                                                  ⁢                            13                                                                                                                                                                                        r                                                      j                            ⁢                                                                                                                  ⁢                            21                                                                                                                                                r                                                      j                            ⁢                                                                                                                  ⁢                            22                                                                                                                                                r                                                      j                            ⁢                                                                                                                  ⁢                            23                                                                                                                                                                                        r                                                      j                            ⁢                                                                                                                  ⁢                            31                                                                                                                                                r                                                      j                            ⁢                                                                                                                  ⁢                            32                                                                                                                                                r                                                      j                            ⁢                                                                                                                  ⁢                            33                                                                                                                                )                                            ,                                                          (        8        )            
Expand Equation (7) to obtain
                              (                                                                      x                  oij                                                                                                      y                  oij                                                                                                      z                  oij                                                              )                =                  (                                                                                                                r                                              j                        ⁢                                                                                                  ⁢                        11                                                              ⁡                                          (                                                                        X                          i                                                -                                                  h                          xj                                                                    )                                                        +                                                            r                                              j                        ⁢                                                                                                  ⁢                        12                                                              ⁡                                          (                                                                        Y                          i                                                -                                                  h                          yj                                                                    )                                                        +                                                            r                                              j                        ⁢                                                                                                  ⁢                        13                                                              ⁡                                          (                                                                        Z                          i                                                -                                                  h                          zj                                                                    )                                                                                                                                                                                      r                                              j                        ⁢                                                                                                  ⁢                        21                                                              ⁡                                          (                                                                        X                          i                                                -                                                  h                          xj                                                                    )                                                        +                                                            r                                              j                        ⁢                                                                                                  ⁢                        22                                                              ⁡                                          (                                                                        Y                          i                                                -                                                  h                          yj                                                                    )                                                        +                                                            r                                              j                        ⁢                                                                                                  ⁢                        23                                                              ⁡                                          (                                                                        Z                          i                                                -                                                  h                          zj                                                                    )                                                                                                                                                                                      r                                              j                        ⁢                                                                                                  ⁢                        31                                                              ⁡                                          (                                                                        X                          i                                                -                                                  h                          xj                                                                    )                                                        +                                                            r                                              j                        ⁢                                                                                                  ⁢                        32                                                              ⁡                                          (                                                                        Y                          i                                                -                                                  h                          yj                                                                    )                                                        +                                                            r                                              j                        ⁢                                                                                                  ⁢                        33                                                              ⁡                                          (                                                                        Z                          i                                                -                                                  h                          zj                                                                    )                                                                                                    )                                    (        9        )            
Substitute Yoij and zoij of Equation (9) into Equation (3) to obtain(fjrj21+rj31ysij)Xi+(fjrj22+rj32ysij)Yi+(fjrj23+rj33ysij)Zi=(fjrj21+rj31ysij)hxj+(fjrj22+rj32ysij)hyj+(fjrj23+rj33ysij)hzj+fjysij  (10)where, 1≦i≦N, 1≦j≦M, and N is the number of point light sources and M is the number of 1D optical lenses.
For N point light sources situated at (Xi,Yi,Zi), 3N independent equations are required for solving the coordinates (Xi,Yi,Zi) of all of N point light sources. Therefore, at least three 1D optical lenses (M=3) are required and installed in the proper focusing directions to satisfy the conditions of the 3N independent equations. However, if the arranged positions of the plurality of point light sources as shown in FIGS. 1(a) and 1(b) are conjugated, a superimposition will occur. Therefore, the condition of the 3N independent equations cannot be satisfied, and the coordinates of N point light sources cannot be obtained. As a result, the effect of the foregoing patented technology cannot be achieved.
For an independent solution of Equation (10), we have to avoid the aforementioned image superimposition. In other words, for N freely moving point light sources, the coordinates (Xi,Yi,Zi) of each point light source can be calculated only when N independent and recognizable images ysij are obtained from each 1D optical lens Lj. For multiple point light sources arranged in a specific position or movement, a specific arrangement for the directions of the 1D optical lenses, or increasing the number of 1D optical lenses is an effective way to obtain the coordinates of multiple point light sources. However it is not a good solution for multiple freely moving point light sources which may easily cause the issue of the image superimposition. According to Equations (9) and (3), the image superimposition can be eliminated if the value rj(Θ,Φ,θj,ρj) is varied properly. In other words, the relation of the coordinate transformed is changed to remove the image superimposition.